Mathematics
AC is diameter, AE is parallel to BC and ∠BAC = 50°.
Statement (1) : ∠EDC + 50° = 180°.
Statement (2) : ∠EDC + ∠EAC = 180°.
Both statements are true.
Both statements are false.
Statement 1 is true, and statement 2 is false.
Statement 1 is false, and statement 2 is true.
Answer
It is given that AC is diameter and angles in a semicircle is a right angle.
⇒ ∠ABC = 90°
Since, AE is parallel to BC and AB is transversal.
⇒ ∠ABC + ∠BAE = 180° [The sum of co-interior angles formed when a transversal intersects two parallel lines is always 180°]
⇒ 90° + ∠BAE = 180°
⇒ ∠BAE = 180° - 90°
⇒ ∠BAE = 90°
⇒ ∠BAC + ∠EAC = 90°
⇒ 50° + ∠EAC = 90°
⇒ ∠EAC = 90° - 50°
⇒ ∠EAC = 40°
AEDC form a cyclic quadrilateral and sum of either pair of opposite angles of a cyclic quadrilateral is 180°.
⇒ ∠EDC + ∠EAC = 180°
⇒ ∠EDC + 40° = 180°
So, statement 1 is false but statement 2 is true.
Hence, option 4 is the correct option.
Related Questions
A circle with center at point O and ∠AOC = 160°.
Statement (1) : Angle x = 100° and angle y = 80°.
Statement (2) : The angle, which an arc of a circle subtends at the center of the circle is double the angle which it subtends at any point on the remaining part of the circumference.
Both statements are true.
Both statements are false.
Statement 1 is true, and statement 2 is false.
Statement 1 is false, and statement 2 is true.
O is the center of the circle, OB = BC and ∠BOC = 20°.
Statement (1) : x = 2 x 20° = 40°
Statement (2) : ∠BOC = 20°.
x = ∠OAB + 20° = ∠OBA + 20° = 40° + 20° = 60°
Both statements are true.
Both statements are false.
Statement 1 is true, and statement 2 is false.
Statement 1 is false, and statement 2 is true.
O is the center of the circle and ∠AOC = 120°.
Statement (1) : ∠ABC = 120°
Statement (2) : ∠ABC + ∠ADC = 180° ⇒ ∠ABC + 60° = 180°.
Both statements are true.
Both statements are false.
Statement 1 is true, and statement 2 is false.
Statement 1 is false, and statement 2 is true.